Abstract
In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite–rank approximations by linear systems. The main result for this approach is to establish correctness of the finite–rank linear system under consideration. And the main result of the paper is to prove correctness of the finite–rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non–zero.
Highlights
Introduction and problem statementFor a long period the main instrument for approximations was based on expansions by complete orthogonal systems
In this paper we consider interpolation problem connected with series by integer shifts of Gaussians
The main result of the paper is to prove correctness of the finite–rank linear system approximation
Summary
For a long period the main instrument for approximations was based on expansions by complete orthogonal systems. Nowadays in different fields of mathematics and applications occurred more and more problems solutions of which needs expansions by incomplete, overdetermined or non-orthogonal systems. Such problems are widely considered for electric or optic signals, filtration, holography, tomography and medicine. We have such examples of overdetermined systems as frames, and non-orthogonal as wavelets, Gabor systems and coherent states, Rvachev’s systems and so on. Let consider approximations by special type series in integer shifts of Gaussians (quadratic exponents with parameters). This system is incomplete in standard spaces but all the same very effective. For the history of this class of approximations, basic results and multiple applications see [3,4,5,6,7,8]
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